STOCHASTIC COAGULATION-FRAGMENTATION PROCESSES WITH A FINITE NUMBER OF PARTICLES AND APPLICATIONS

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1 Submitted to the Annals of Applied Probability STOCHASTIC COAGULATION-FRAGMENTATION PROCESSES WITH A FINITE NUMBER OF PARTICLES AND APPLICATIONS By Nathanael Hoze and David Holcman,, Institute of Integrative Biology, ETH Zurich, Zurich, Switzerland Newton Institute and Department of Applied Mathematics and Theoretical Physics (DAMPT), University of Cambridge, CB30DS United Kingdom and Ecole Normale Supérieure, 46 rue d Ulm Paris, France Coagulation-fragmentation processes describe the stochastic association and dissociation of particles in clusters Cluster dynamics with cluster-cluster interactions for a finite number of particles has recently attracted attention especially in stochastic analysis and statistical physics of cellular biology, as novel experimental data are now available, but their interpretation remains challenging We derive here probability distribution functions for clusters that can either aggregate upon binding to form clusters of arbitrary sizes or a single cluster can dissociate into two sub-clusters Using combinatorics properties and Markov chain representation, we compute steady-state distributions and moments for the number of particles per cluster in the case where the coagulation and fragmentation rates follow a detailed balance condition We obtain explicit and asymptotic formulas for the cluster size and the number of clusters in terms of hypergeometric functions To further characterize clustering, we introduce and discuss two mean times: one is the mean time two particles spend together before they separate and the other is the mean time they spend separated before they meet again for the first time Finally we discuss applications of the present stochastic coagulation-fragmentation framework in cell biology Introduction Clustering appears in various areas of science such as astrophysics, where masses can form aggregate under gravitation, biochemistry where molecules have to meet to react, colloids that aggregate in solution or ecology where prey-predator have to meet to stabilize populations A century ago, Von Smoluchowski [29] described an irreversible aggregation of many particles in clusters Later on, a set of coagulation-fragmentation equations was proposed by Becker-Döring when clusters can lose or gain only one particle at a time [6, 7, 2, 33] Nowadays, continuous limit analysis [3, 26], David Holcman is supported by a Marie Curie Award and a Simons fellowship He thanks the Institute of Mathematics, Oxford University, for hospitality MSC 200 subject classifications: Primary 60J20, Markov chain, Coagulationfragmentation processes, Stochastic processes; secondary 05A7

2 2 N HOZE AND D HOLCMAN determinist, stochastic, asymptotical and numerical methods are developed to study clustering and estimate the number of clusters [2, 9, 32, 27, 8] and their sizes The coagulation models mentioned above have been extended for cluster growth with a finite number of particles [23, 24], known as Marcus- Lushnikov process, and more recently for discrete coagulation-fragmentation models with an infinite number of particles [5] Many statistical and probabilistic studies have been made on the Marcus-Lushnikov process [8], however, much less is known about the statistical properties for the coagulationfragmentation of a finite number of particles [3] Of particular interest in stochastic biology are models of coagulation-fragmentation where the cluster size cannot exceed a given threshold [4, 34] These models are used in genetics to describe the organization in clusters of the chromosome ends [4] or to model viral capsid assembly in cells [35, 6, 7] This article presents several exact and asymptotic results and it aims to attract attention toward developments of applied probability with direct applications to interpret data We recall that the Smoluchowski equations for coagulation-fragmentation consist of an infinite system of differential equations for the number n j (t) of clusters of size j at time t in a population of infinite size [29] The index j can take values between and and () dn j (t) dt j A(k, j k)n k (t)n j k (t) n j (t) A(j, k)n k (t) 2 k0 k k 2 n j j(t) B(k, j k) + B(j, k)n k+j (t), k where the first line in the right-hand side corresponds to the coagulation and the second accounts for the fragmentation The coagulation kernel A(i, j) is the rate at which two clusters of size i and j coalesce to form a cluster of size i + j, while the fragmentation kernel B(i, j) is the rate at which a cluster of size i + j dissociates into a cluster of size i and a cluster of size j When cluster sizes cannot exceed a threshold M, the system of equations () is truncated and the coagulation kernel is A(i, j) 0 if i + j M [36, 37] This system of equations is a mean-field deterministic model of the coagulation-fragmentation process that describes a discrete number of particles aggregating and dissociating, but it does not allow a complete analysis of the cluster distribution in the small size limit [3] The goal of this paper is to study a stochastic system of coagulationfragmentation with a limited number of particles We present exact soluimsart-aap ver 204/0/6 file: HozeCoagulation_imsAPRIL207-dhnhdhtex date: May, 207

3 STOCHASTIC COAGULATION-FRAGMENTATION 3 tions and expressions for the distribution, statistical moments of clusters and number of particles per cluster, using the following generic rules of coagulation-fragmentation: a cluster of size i + j can give rise to two clusters of size i and j at a rate F (i, j), and two clusters of size i and j form a new cluster of size i + j at a rate C(i, j) We focus our analysis on coagulation-fragmentation processes (CFP) that verify the detailed balance condition [2], for which there exists a function a(i) a i such that i, j N (2) C(i, j) a(i + j) F (i, j) a(i)a(j) The detailed balance condition ensures that, at equilibrium, each elementary coagulation or fragmentation process is equilibrated by its reciprocal process Using exact formulas for probabilities for these configurations when the total number of clusters is fixed, we compute the probability distribution function of the number of clusters We also compute the probability distribution that the number of cluster of size i is m i so that the distribution of sizes of the ensemble of clusters is (m,, m n ) When there are exactly N particles and the total number of clusters is fixed to K, we have the following identity for number conservation (3) N m i K i We shall show that when the total number of clusters is K, the conditional probability distribution function is given by (4) p (m,, m N K) a() m a(n) mn, m!m N! where the normalization constant can be computed explicitly (see formula 26) We will use this formula to compute the statistical moments for the cluster distributions Using a combinatorial approach, we present several explicit formula for the steady-state distributions of clusters Some of the results presented here were previously announced without proofs in the short letter [4] The paper is organized as follows In section, we present the stochastic model of coagulation-fragmentation for N independent particles, that we analyze by a Markov chain We obtain explicit formula for the cluster configuration combinatorics using the partition of the integer N, for the distribution of particles in clusters We derive the time evolution equations

4 4 N HOZE AND D HOLCMAN for the number of clusters These equations represent a novel Markov chain, which allows us to determine the number of clusters at steady-state By combining the expression for number of clusters with the distribution of cluster in configuration conditioned on the number of clusters, we obtain the distribution of the particles in clusters In section 3, we introduce two characteristic times for studying the time distribution two particles are in a cluster These two times characterize the dynamics of exchange of particles between clusters: the first one is the mean time that two particles spend together before they separate, and the second is the mean time that they spend separated before they meet again for the first time The colocalization probability of two particles is defined as the fraction of time, the particles spend together Sections 4-6 provide direct applications of these results to specific CFPs: we focus specifically on the case of constant coagulation and fragmentation kernels in section 4, for which we obtain analytical expressions for the clusters distributions We also obtain several formula when the cluster size is limited In this article, to determine statistics of the cluster distribution, we alternatively study two different systems First, we study the distribution of clusters using the integer partition of the total number of particles We obtain the probability distribution of cluster configurations In this process, we use the term coagulation when two clusters of a given size coalesce and form a new cluster We use the term fragmentation to describe the separation of a cluster into two smaller ones The coagulation and fragmentation kernels that we have chosen here allow us to perform another analysis: when the number of clusters is fixed, the overall rates of coagulation and fragmentation are independent of the configurations of the clusters We will study aggregation-fragmentation, where the number of clusters is known In that case, we shall use the following terminology: formation describes the change when a distribution of K becomes K clusters We use separation to describe the process by which a distribution of K clusters is transformed into a distribution of K + clusters Coagulation-fragmentation with a finite number of independent particles Stochastic coagulation-fragmentation equations for a finite number of particles To describe the steady-state distribution for a CFP stochastic model with a finite number of N particles, we shall use a continuous-time Markov chain in the space of cluster configurations The N particles distributed in clusters of size (n,, n N ) can undergo coagulation or fragimsart-aap ver 204/0/6 file: HozeCoagulation_imsAPRIL207-dhnhdhtex date: May, 207

5 STOCHASTIC COAGULATION-FRAGMENTATION 5 mentation events under the constraint that (5) N n k N k To study the distribution of particles in clusters, we use the decomposition of the integer N as the sum of positive integers (integer partition) [4] The partitions of the integer N are described in N dimensions by the ensemble (6) P N {(n,, n N ) N N ; N n i N and n n N 0} i The probability P (n,, n N, t) of the configuration (n,, n N ) at time t satisfies an ensemble of closed equations Indeed, by considering all the possible coagulation or fragmentation events, the Master equation is obtained by considering the events occurring between time t and t + t: Two clusters of size n i and n j coagulate with a probability C(n i, n j ) t to form a cluster of size n i + n j A cluster of size n i dissociates into two clusters of size k and n i k with a probability F (k, n i k) t Nothing happens with the probability N N i ji+ C(n i, n j ) t N ni i k F (k, n i k) t Thus, the Master equations are N d dt P (n,, n N, t) + N N i ji+ k n i >0,n j >0 n i +n j n k C(n i, n j ) + N i n i k F (k, n i k) P (n,, n N, t) C(n i, n j)p (n,, n i,, n j,, n N, t) (7) + N N i ji+ F (n i, n j )P (n,, n i + n j,, n N, t) Moreover, C(n i, n j ) 0 if either n i or n j is equal to 0 We now introduce an ensemble in N N which consists of integer decompositions of clusters with a given size: (8) N P N {(m,, m N ) N N ; im i N and m,, m N 0} i

6 6 N HOZE AND D HOLCMAN In the ensemble P N, m i is the number of occurrence of the integer i in the partition of the integer NThe two ensembles P N and P N correspond to different representations of the clusters distributions For example, when there are N 9 particles, distributed in two clusters of one particle, two clusters of two, and one cluster of three Then the distributions are (3, 2, 2,,, 0, 0, 0, 0) P 9, and (2, 2,, 0, 0, 0, 0, 0, 0) P 9 A sufficient condition to obtain an invariant measure of the steady state probability is the reversibility of the CFP [20, 22] where the coagulationfragmentation kernel satisfies the detailed balance condition: there exists a function a(i) a i such that [2] (9) C(i, j)a i a j F (i, j)a i+j The functions a i characterize the ratio of the coagulation and fragmentation rates, and any function of the form a i αi a i with α 0 satisfies the above condition This condition insures the reversibility of the Markov chain and guarantees the existence of an invariant measure [2], where the steady-state probability of a given configuration (m,, m N ) P N is (0) P (m,, m N ) a m am N N C N m!m N!, where C N is a normalization constant An explicit computation of the normalization constant is difficult [3] Here, we propose to estimate the probability of occurrence of a certain cluster configuration (m,, m N ) by limiting the study to the configurations of a given number of particles At this stage, we shall explain the rational for computing expression (0) This expression is the distribution at equilibrium of particles in clusters, where the dissociation (resp association) rate is proportional to the number of elements (minus one) (resp the number of pairs of particles) The equilibrium probability distributions associated to the Markov chain configuration (m,, m N ) is computed from analyzing the transition between the two neighboring states (m,, m i,, m j,, m i+j +,, m N ) and (m,, m N ) It is obtained first from the coagulation rate ψ(i, j) of a cluster of size i with one of size j, given the distribution (m,, m N ) The rate ψ(i, j) is given by () (2) ψ(i, j) 2 C(i, j)m im j if i j C(i, i)m i (m i ) otherwise The factor 2 accounts for the symmetric cases ψ(i, j) and ψ(j, i) The fragmentation rate ϕ(i, j) from i+j to (i, j), that accounts for the transition

7 STOCHASTIC COAGULATION-FRAGMENTATION 7 from the configuration (m,, m i,, m j,, m i+j +,, m N ) to (m,, m N ) (Fig ) is (3) ϕ(i, j) F (i, j)(m i+j + ) Thus, the stationary probability π satisfies the relation π (m,, m i, m j, m i+j +,, m N ) π (m,, m N ) (4) ψ(i, j) ϕ(i, j) C(i, j) m i m j 2 F (i, j) m i+j + a i a j m i m j 2 m i+j + a i+j A direct computation shows that the probability P (m,, m N ) defined in eq (0) satisfies equation (4) (m,, m N ) (m,, m i k +,, m k +,, m i,, m N ) Fragmentation F (k, i k)m i rkov chain representation of the transitions Starting with a configuration N ), m i is the number of clusters of size i and N is the total number of particles entation rate of a cluster of size i into one cluster of size k and one of size i k k)m i, while the rate of formation of a cluster of size i + j from two clusters of j is equal to C(i, j)m i m j luster partitions with a finite number of particles To determine er distribution at equilibrium, we compute here the probability of ration when the number of clusters K is fixed We also find the ity of having K clusters The number of distributions of N particles lusters is the cardinal of the ensemble {(n,, n K ) N K ; K n i N and n n K }, i p ver 204/0/6 file: HozeCoagulation_imsAPRIL207-dhnhdhtex date: May, 207

8 8 N HOZE AND D HOLCMAN which is also the ensemble of the partitions of the integer N as a sum of K integers This ensemble is in bijection with N (6) P N,K {(m,, m N ) N N ; im i N and i N m i K}, i where the transformation P N,K P N,K defined by K K (7) (n,, n K ) (m,, m N ) ( {ni },, i i {ni N}) maps the partition (n,, n K ), where N is written as a sum of K positive integers, to the number of occurrence of each integer into the image partition The partitions of N are written as (8) P N K P N,K and P N K P N,K In sections 4, 5 and 6, we derive explicitly expressions for the probabilities of configurations in P N,K 3 Statistical moments for the cluster configurations when the number of clusters is fixed We show now that the probability of configuration (m,, m N ), when the total number of clusters is equal to K, is given by (9) where (20) p (m,, m N K) (m i ) P N,K a m am N N m!m N!, a m am N N m!m N! The normalization factor of eq (9) is computed using the following result: Remark We consider the functions (2) S(x) a i x i i

9 and the partial sums STOCHASTIC COAGULATION-FRAGMENTATION 9 (22) S N (x) N a i x i i The K th power of these functions, that is S K and SN K have the same N th order coefficient and this coefficient determines Moreover, the function (23) g(x, y) exp(s(x)y) is a generating function of the Proof:The number of configurations (m,, m N ) that satisfy the conditions i m i K and i im i N is the N th -order coefficient of the multinomial expansion (a x + + a N x N ) K (24) (m,,m N ), m i K K! m! m N! (a x ) m (a N x N ) m N Using the N-tuple (x, x 2,, x N ) (X, X 2,, X N ), we obtain (25) (a X + + a N X N ) K K! (m,,m N ), m i K i a m i i m i! X i im i We can group the terms by the exponents of X, which are equal to i im i In particular, for a partition (m,, m N ) P N,K, the exponent is i im i N, and the N th order coefficient in expression (25) is equal to a m i i (26) m! m N! More generally, (m,,m N ) P N,K (27) S K (X) K! nk (m,,m N ) P n,k a m i i m! m N! Xn (28) nk (m,,m N ) P n,k C n,k X n

10 0 N HOZE AND D HOLCMAN It follows that the function defined by (29) (30) (3) is a generating function of the g(x, Y ) exp(s(x)y ) S K (X) Y K K! K0 C n,k X n Y K K0 nk Remark 2 The coefficients defined by relation (20) satisfy the induction formula: (32) with (33) (N + )C N+,K N K+ k0 C N,N an N! C N, a N (k + )a k+ C N k,k Proof:We start with the formulas (33) The coefficient C N,N is obtained from the unique partition in P N,N, which is given by m N and m i 0 if i > Therefore, C N,N an N! The coefficient C N, is obtained from the partition m N and m i 0 if i < N, and thus C N, a N We now prove relation (32) Differentiating the function σ(x) SK (x) K!, we get (34) σ (x) S (x) SK (x) (K )! We evaluate the lhs of (34) using eq (28), and obtain ( ) σ (x) C n,k x n (35) nk nk (n + )C n+,k x n x K (n + K)C n+k,k x n n0

11 STOCHASTIC COAGULATION-FRAGMENTATION We evaluate the rhs of (34) using the definition of S (2) and we obtain ( ) ( ) σ (x) (i + )a i+ x i C i,k x i (36) i0 ik ( n ) x K (k + )a k+ C n k+k,k x n n0 k0 Thus, by equalizing the N th order coefficient of σ (x), we have (37) (N + )C N+,K N K+ k0 (k + )a k+ C N k,k Next, we estimate various moments when the number of clusters is fixed We summarize the main result in the Theorem When the number of clusters is equal to K for a total of N particles, the mean number of clusters of size i is (38) M i N,K a i C N i,k, where a i and are defined in (9) and (20) respectively Proof:The mean number of clusters of size i when the total number of clusters is K is given by the following sum (39) (40) M i N,K P N,K m i p (m,, m N ) P N,K P N,K,m i>0 a m m am N N i m!m N! a m a am i i a m N N i m!(m i )!m N!, where the subscript P N,K, m i > 0 in the sum characterizes the partitions in P N,K containing at least one occurrence of the integer i By considering the partitions of P N,K where i appears at least once and removing one i, we obtain exactly the partitions of P N i,k, except for the number of occurrence of i, where the corresponding partitions in both sets have the same number of repetitions, ie (4) j i, m j in (m,, m N i ) P N i,k

12 2 N HOZE AND D HOLCMAN is equal to (42) m j in (m,, m N ) P N,K and m i in (m,, m N i ) P N i,k is equal to m i + in (m,, m N ) P N,K There are no clusters larger than N i in the partitions (m j 0 for j > N i for (m,, m N ) P N,K, m i > 0) Thus M i N,K a i P N,K,m i>0 m a P N i,k a m a am i i a m N N i m!(m i )!m N! am i i a m N N m!m i!m N! (43) We further have: C N i,k a i Remark 3 M i N,K 0 if i > N K + Proof:When the N particles are distributed in K clusters, the largest cluster contains at most N K + particles The corresponding partition in P N,K is given by m K, m N K+, and m j 0 otherwise In the following, we determine the second moment of the number of clusters of size i (44) M 2 i N,K m 2 i P N,K a m am N N m!m N! and the covariance of the number of clusters of size i and j (45) M 2 i,j N,K P N,K a m m i m am N N j m!m N! Theorem 2 The second moment of the number of clusters of size i is (46) and the covariance is M 2 i N,K a 2 i C N 2i,K 2 + a i C N i,k, (47) M 2 i,j N,K a i a j C N i j,k 2

13 STOCHASTIC COAGULATION-FRAGMENTATION 3 Proof:The variance of the number of clusters of size i is given by Mi 2 N,K (48) m i P N,K m i P N,K m 2 i m i P N,K,m i> a m am N N m!m N! [m i (m i ) + m i ] am am N N m!m N! a 2 i a m a m i 2 i a m N N m! (m i 2)! m N! + M i N,K Using an argument similar to the proof of Theorem, we obtain M 2 i N,K m i P N 2i,K 2 a 2 i a m a m i i a m N N m! m i! m N! + M i N,K (49) a 2 i C N 2i,K 2 + a i C N i,k The covariance of the number of clusters of size i and j is obtained from the term (50) M 2 i,j N,K P N,K which, by the same reasoning, leads to a m m i m am N N j m!m N!, (5) M 2 i,j N,K a i a j C N i j,k 2 2 Distribution of the number of clusters In the previous section, we computed the probability distribution of a cluster configuration and determined the statistical moments for a fixed number of clusters In this section, we study the statistics of the entire cluster configurations We focus here on the probability distribution of the number of clusters and we shall compute the time dependent probability density function (52) P K (t) P {K clusters at time t},

14 4 N HOZE AND D HOLCMAN which is a birth-and-death process that we investigate using a Markov chain The probability of having K clusters at time t + t is the sum of the probability of starting at time t with K clusters and one of them dissociates into two smaller ones plus the probability of starting with K + clusters and two of them associate plus the probability of starting with K and nothing happens (Fig 2) The first probability is the product of P K by the f N f N- f N-2 f N-3 N N- N-2 N-3 c N- c N-2 c N-3 c N-4 Fig 2 Markov chain representation for the number of clusters s K (respectively f K) is the separation (respectively formation) rate of a cluster when there are K clusters transition rate s K t to go from state with K clusters to K, while the second is the transition from K + to K, which is the product of P K+ by the transition rate f K+ t of going from K + clusters to K Thus the master equations are given by (53) P (t) P K (t) P N (t) s P (t) + f 2 P 2 (t) (f K + s K )P K (t) + f K+ P K+ (t) + s K P K (t) f N P N (t) + s N P N (t) In section 4, 5 and 6, we will solve this system of equations explicitly at steady-state for particular formation and separation kernels We now derive a general formula for the steady state probability (54) Π K lim t P K (t) of having K clusters at steady state and express it in terms of the a i The steady state probabilities of the number of clusters are solution of the system (55) 0 s Π + f 2 Π 2 0 (f K + s K )Π K + f K+ Π K+ + s K Π K 0 f N Π N + s N Π N,

15 STOCHASTIC COAGULATION-FRAGMENTATION 5 with the normalization condition N (56) Π K K The probabilities Π K are given by the ratio (57) Π K Π K s K f K for K 2 and the coefficients s K and f K are the mean-field separation and formation rates respectively Whereas the cluster configurations when the number of clusters is fixed depend only on the kernel a i, the statistics of the number of clusters depend on the cluster fragmentation and coagulation rates F and C In the following, we will focus on the coagulation condition C(i, j) and the fragmentation F (i, j) a ia j a i+j to state the Theorem 2 When C(i, j) and F (i, j) a ia j a i+j, the separation rate when there are K clusters is given by N i i j s K a ja i j C N i,k (58) and the formation rate when there are K clusters is K(K ) (59) f K 2 Proof:The total dissociation rate d(n) of a cluster of size n is obtained by summing over all the possible sizes resulting from the dissociation and is given by (60) d(n) n F (i, n i) i n i a i a n i a n The rate at which a given configuration (m,, m N ) P N,K dissociates is N i d(i)m i The separation rate for K clusters is thus (6) s K N d(i)m i p (m,, m N ) P N,K i N d(i) M i N,K i

16 6 N HOZE AND D HOLCMAN Using relation (43), the separation rate becomes (62) s K N i d(i)a ic N i,k The separation rate of a distribution of K clusters eq (62) can thus be expressed as a function of the (63) s K N i i j a ja i j C N i,k For K, the only cluster is of size N and the separation rate is (64) s d(n) The formation rates are given by f K N m i (m i )C(i, i) + 2 P N,K i i j (65) m i m j C(i, j) p (m,, m N K) For a distribution of K clusters, this is equal to the number of cluster pairs (66) f K K(K ) 2 We are now in position to study the statistics of the entire cluster configurations Indeed, using Bayes rule, the probability of a configuration (m,, m N ), that contains K clusters is the product of the conditional probability p (m,, m N K) by the probability of having K clusters (67) p (m,, m N, K) p (m,, m N K)Π K The mean number of clusters of size i is thus (68) N M i N Π K M i N,K K 3 Invariant of clusters dynamics We introduce and compute here several measures of the cluster configurations that appear when the system is in a global steady state First, we compute the probability to find two particles in the same cluster, and second we measure two time scales associated to particle dynamics in an ensemble of clusters: ) the mean time that

17 STOCHASTIC COAGULATION-FRAGMENTATION 7 two particles spend together before they separate and 2) the mean time that they spend separated before they meet again for the first time (Fig 3) The probability that two given particles are together is of interest in several cell biology examples: for instance, some genes can be silenced if the telomeres that carry them are forming a cluster [38] Time T S Time T R Fig 3 Cluster formation and dissociation The time to separation T S is the mean time for two specific particles (black) to spend in the same cluster Before physical separation, the cluster can aggregate with other clusters (grey) or some particles except either of the two can be separated The time to association T R is the mean time the two particles meet again after separation for the first time 3 The probability to find two particles in the same cluster When the mean number of clusters has reached its equilibrium, particles can still be exchanged between clusters To characterize this exchange, we compute the probability to find two particles in the same cluster When the distribution of the clusters is (n,, n N ), the probability P 2 (n,, n N ) to find two given particles in the same cluster is obtained by using the probability to choose the first particle in the cluster n i, which is equal to the number of particles in the cluster divided by the total number of particles n i N The probability to have the second particle in the same cluster is n i N Summing over all possibilities, we get (69) P 2 (n,, n N ) K n i n i N N K N(N ) ( n 2 i N) i i This probability is similar to Simpson s diversity index [28], a measure frequently used to quantify the diversity of ecosystems We note that (70) N n 2 j j N i 2 m i i

18 8 N HOZE AND D HOLCMAN Thus, when the distribution (n,, n N ) contains K clusters, we use (n,, n K ) P N,K and obtain by summing over all configurations of K clusters K (7) p(n,, n K K) n 2 j (n,,n K ) P N,K j (72) (73) N p (m i ) j 2 m j (m i ) P N,K j N j 2 j (m i ) P N,K N j 2 M j N,K, j m j p (m i ) where M j N,K is the mean number of clusters of size j, when there are N particles distributed in K clusters (eq (38)) Taking into account all possible distributions of clusters, we obtain that the probability P 2 to find two particles in the same cluster is (74) N P 2 K (n,,n K ) P N,K P 2 (n,, n K )p(n i )Π K, which can be written, using expressions (69) and (73) as (75) P 2 N(N ) N N Π K K j j 2 M j N,K N This approach can be generalized to the probability of having n 2 particles together 32 Mean time for two particles to stay together in a cluster We define the mean time to separation (MTS) as the mean time between the arrival of two given particles in a cluster and their separation after dissociation of the cluster We first compute the probability of a configuration (m,, m N ) conditioned on having two particles in the same cluster We then derive the transition rates from this particular configuration to any of the states accessible by a single dissociation or association event (see Fig ) The accessible states are divided into two classes: first, the configurations for which the particles are not initially in the same cluster (separated state) and second the ones where they are together The current state is the configuration (m,, m N ), for which the particles are in the

19 STOCHASTIC COAGULATION-FRAGMENTATION 9 same cluster Upon a coagulation event the particles stay in the same cluster A dissociation event can either occur for a cluster that did not contain the tracking particles or for the cluster that contained the particles In the latter case, there are two possibilities: either the particles stay together after dissociation, or they are redistributed to two different clusters (separated state) The rate of dissociation from the ensemble (m,, m N ) to (m,, m i +,, m k i +,, m k,, m N ) is 2F (i, k i)m k if i k/2 and F ( k 2, k 2 )m k otherwise Proposition The probability that two particles in a cluster of size k, given a configuration (m,, m N ) P N,K separate after a dissociation is k p S (k; m,, m N ) k(k ) N i d(i)m i(2k i)f (i, k i) i (76) Proof:The probability that two particles in a cluster of size k separates after any dissociation event is equal to the product of the probability that the particles are in the cluster multiplied by the probability that this dissociation results in the effective separation of the particles The first probability is obtained by considering the total dissociation rate d(n) of a cluster of size n (see eq (60)) The probability that the cluster containing the two particles effectively dissociates is thus proportional to d(k) and is normalized by the total dissociation rate for the cluster configuration (m,, m N ) : (77) d(k) N i d(i)m i The probability of an effective separation P sep is the complementary of the probability that the two particles stay together When the two resulting clusters are of size i and k i, this probability is equal to (78) P sep i i(i ) + (k i)(k i ) k(k ) The probability that the two particles initially in a cluster of size k will be separated is thus equal to p S (k) (79) d(k) K i d(i)m i k i ( ) i(i ) + (k i)(k i ) F (i, k i) k(k ) d(k) k k(k ) N i d(i)m F (i, k i)i(2k i) i i

20 20 N HOZE AND D HOLCMAN The transition probability from the state (m,, m N ) to the separated one equals the sum of the probabilities p S (k) to be separated over all cluster sizes k, (80) P S (m,, m N ) K p S (k)m k To derive the MTS, we first write the transition matrix of the Markov chain representing the transitions between the configurations of separated and non-separated states, and second, we determine the mean transition times between the states Ordering the configurations (m,, m N ) by arbitrary indices I, we define the transition matrix T of size (q(n) + ) (q(n) + ), where q(n) is the total number of different partitions of the integer N The elements of T of the first q(n) rows and columns are the transition probabilities between states where the two particles are together, while index q(n) + represents the separated state For I, J (,, q(n)), the matrix entries [T ] I,J are either the coagulation or dissociation rates given above (see Fig ), while the transitions [T ] I,q(N)+ are equal to the rates to the separated state Because the separated state is absorbing, we finally set [T ] q(n)+,q(n)+ Additionally, the matrix is normalized into a stochastic matrix such that (8) q(n)+ I, J k [T ] I,J We now estimate the mean time the two particles stay together The mean time τ(m,, m N ) from a configuration (m,, m N ) to any configuration (m,, m N ) accessible by a single coagulation or fragmentation event is the reciprocal of the sum of all transition rates (82) ( N τ(m,, m N ) d(k)m k + k ) K(K ), 2 where K N i m i and the coagulation kernel is constant C(i, j) We represent the transition times in a vector τ τ(n, 0,, 0) τ(n 2,,, 0) (83) τ τ(0,, 0, )

21 STOCHASTIC COAGULATION-FRAGMENTATION 2 We shall now compute the vector t(n, 0,, 0) t(n 2,,, 0) (84) t t(0,, 0, ) which is the MTS for an ensemble of cluster configuration (m,, m N ): it is related to the vector τ by [25] (85) t T n τ n0 The vector t is computed using the matrices (86) A I q(n)+ T and (87) A [ A q(n)+ ], where A q(n)+ is the matrix A from which the q(n) + row and column were removed Equation (85) is equivalent to (88) t A τ The MTS averaged over the equilibrium configuration distribution is (89) T S p T t where p T [p (m,, m N )] (m,,m N ) and p (m,, m N ) is the probability distribution of the configuration (m,, m N ) when the two particles are in the same cluster Using Bayes theorem, the probabilities p (m,, m N ) are given by (90) p (m,, m N ) P 2(m,, m N )p(m,, m N ), P 2 where we recall that p(m,, m N ) is the probability of the configuration (m,, m N ), P 2 is the probability that two specific particles are in the same cluster (see computation eq (75)), and P 2 (m,, m N ) is the probability that the two particles are in the same cluster configuration (m,, m N ) The conditional probability p (m,, m N ) to select two particles in the same cluster, conditioned on the distribution (m,, m N ) is larger for clusters of larger sizes We conclude this section with the following

22 22 N HOZE AND D HOLCMAN Remark 4 The time that two particles spend separated T R can be similarly determined, only the absorbing state in the transition matrix is the state at which the two particles are in the same cluster Interestingly, the probability two particles are together is the fraction of time they spend together and is given by the ratio (9) P 2 T S T S + T R In the rest of the manuscript, we apply the previous analysis to three examples of coagulation-fragmentation with a finite number of particles 4 Example : the case a i a We consider the case of a constant kernel a i a To compute the separation and formation rates s K and f K, we use that F (i, j) a and C(i, j) This fragmentation kernel corresponds to the following model: a cluster of size n dissociates at a rate n i F (i, n i) (n )a and the sizes of the resulting clusters are uniformly distributed between and n When there are N particles and a total number of clusters K, the partition of clusters is denoted (n,, n K ) P N,K The total transition rate from a configuration of K to K + clusters is the sum over all possible dissociation rates (92) s K K (n i )a (N K)a i The formation rate is proportional to the number of pairs (93) f K K K i ji+ C(n i, n j ) K(K ) 2 Following relation (53), the steady-state probability Π K for the number of clusters of size K satisfies the time independent master equation s Π f 2 Π 2, (94) (f K + s K )Π K f K+ Π K+ + s K Π K, which leads to the relation (95) f N Π N s N Π N, Π K+ (2a) K (N )! K!(K + )!(N K )! Π

23 STOCHASTIC COAGULATION-FRAGMENTATION 23 Using the normalization condition K Π K, the probability Π can be expressed as a hypergeometric series (96) where (97) Π F ( N + ; 2; 2a), F (a; b; z) n0 (a) n z n (b) n n!, is Kummer s confluent hypergeometric function ([] pp ) and (98) (x) n x(x + )(x + n ) is the Pochhammer symbol The average number of clusters at steady state (99) µ (a) N KΠ K K ( ) d Π z F ( N + ; 2; z) dz z 2a The derivative of the Kummer s function is (00) d dz F (a; b; z) a b F (a + ; b + ; z) Finally the mean number of clusters is expressed as (0) µ (a) + a(n ) F ( N + 2; 3; 2a) F ( N + ; 2; 2a), + a(n )G, where we note G the function defined by (02) G F ( N + 2; 3; 2a) F ( N + ; 2; 2a) More generally, we introduce the functions G i defined by (03) G i F ( N + + i; 2 + i; 2a) F ( N + ; 2; 2a) Following the procedure presented above, all moments of the probability distribution Π K can be computed and the n th -order moment µ n is expressed using the operator H defined by (04) H(f)(z) d dz zf(z),

24 24 N HOZE AND D HOLCMAN by (05) µ n N n K n Π K H(n) ( F ( N + ; 2; z)) z 2a F ( N + ; 2; 2a) Using the differentiation formula for the hypergeometric function (00), the moments µ n can be written as (06) (07) µ n n k0 n k0 α n k (N )! (k + )!(N k)! 2k a k G k, αk n Π k+ G k, Π where the coefficients αk n are given by k! k/2 (k + j0 ( )j j)n + (j + ) n (k j)! αk n k! (k )/2 j0 ( ) j (k + j)n (j + ) n (k j)! if k is even, if k is odd, and α n 0 αn n We can thus obtain the variance of the number of clusters, given by V (a) µ 2 µ 2 a(n )G (a, N) a2 (N )(N 2)G 2 (a, N) (08) a 2 (N ) 2 G 2 (a, N) 4 Asymptotic formulas for the mean and variance of the cluster number We provide here approximations for the functions G n defined in eq (03) Kummer s function can be expressed in terms of generalized Laguerre polynomials (09) F ( n; b; z) n! L b n (z) (b) n where n is an integer The asymptotic behavior of Laguerre polynomials for large n, fixed x > 0 and α, is given by [30] L α n( x) n α π e x 2 x α ( exp 2 x(n + α + )( ) + 2 m/2 ν ) C ν (x)n ν/2 + O(n m/2 ),

25 STOCHASTIC COAGULATION-FRAGMENTATION 25 where m is a positive integer and C ν (x) is a regular and bounded functions for x > 0, independent of N Using the leading order term in the expansion for m 2, we get L α n( x) n α π (0) e x 2 x α ( exp 2 x(n + α + )( ) ) + C (x)n /2 + O(n ) 2 We can now evaluate G by using the asymptotic expansion for F ( N + ; 2; 2a) and F ( N + 2; 3; 2a) For large N, we have F ( N + ; 2; 2a) (N ) /4 N 2 e a ( exp 2 )( 2aN + O( ) ), π (2a) 3/4 N () and (N 2) 3/4 F ( N + 2; 3; 2a) N(N ) 2 e a ( exp 2 )( 2a(N /2) + O( ) ) π (2a) 5/4 N (2) Finally G (a, N) (3) 2a N exp 2 an exp ( 2 2a(N /2) 2 )( 2aN + O( ) ), N ( ) a 2N G (a, N) Similarly, the present computation can be generalized to obtain the asymptotic approximation G n for the functions G n, valid for large N and fixed n, ( ) (n + )! a (4) G n (a, N) exp n (2aN) n/2 2N In Figure 4, we compare the exact value of G (defined in 02) with the approximation G given by expression (3) The approximation is more accurate for intermediate values of a (see Fig 4B) In addition, the error function G G G has a discontinuity in the derivative for a 0 Indeed at a 0, the function G and G cross each other and thus G G has a singular derivative, shown by a cusp type behavior in Fig 4B Moreover, the function G is always decreasing with N, however its approximation G

26 26 N HOZE AND D HOLCMAN is non monotonic (Fig 4A) Finally, by using the asymptotic expression (3), we find the approximation of the number of clusters for large N, µ (a) + ( ) a (5) 2aN exp 2N To obtain a numerical approximation of G for small a, we used the finite continued fraction decomposition for the ratio of hypergeometric functions F [9], and obtain (6) G (a, N) F ( N + 2; 3; 2a) F ( N + ; 2; 2a) (N + )a/3 (N 2)a/6 (N + 2)a/0 (N 3)a/5 + a/(n )(2N 3) +a/(n ) We obtain the Taylor expansion of G for small a from the continued fraction, G (a, N) N + ( ) (N + )(N 2) (N + )2 a + + a ( ) (N + )(N 2)(N + 2) (N + )(N 2)2 (N + )3 + + a (7) + o(a 3 ) This expression is computed from the continued fraction G written (8) G (a) + F (a)a, f a + + F 2 (a) where the general term of the sequence F n is (9) F n (a) f n + F n+ (a)a

27 STOCHASTIC COAGULATION-FRAGMENTATION 27 A 0 0 ~ G -G ~ G G, G B N a a0 a00 a, Fig 4 Approximation of G (A) Plot of G (black) and approximation G (red, eq (3)) vs N for a, 0, 00 and, 000 (B) Comparison of the values of G and G, measured as G G G N

28 28 N HOZE AND D HOLCMAN with F n+ (a) O() G can also be written as (20) G (a) af + af af n To obtain a third-order Taylor expansion of G, we simply used F, F 2, F 3 and we have truncated the rest of continued fraction to obtain (2) The expansion of F is (22) G (a) F (a)a + F 2 (a)a 2 F 3 (a)a 3 + o(a 3 ) F (a) f + F 2 (a)a f ( F 2 (a)a + F 2 2 (a)a 2 ) + o(a 2 ), f ( f 2 ( f 3 a)a + f 2 2 (a)a 2 ) + o(a 2 ), f f f 2 a + (f f 2 f 3 + f f 2 2 )a 2 + o(a 2 ) Using F expansion into G, we finally get (23) G (a) f a + (f 2 + f f 2 )a 2 (f f 2 f 3 + f f f 3 )a 3 + o(a 3 ) 42 Statistics for the number of clusters of a given size We now compute several moments for the size of clusters when there are N particles Using relation (38), we first obtain the expression for the mean number of clusters of size n when there are K clusters M n N,K m n p (m i K) (m i ) P N,K (24) a C N n,k Determining this relation requires computing the normalizing constant given in eq (20) The normalizing constant is the Nth order coefficient of S K, where S is the generating function (2) (25) S(x) a i x i a x x i

29 STOCHASTIC COAGULATION-FRAGMENTATION 29 The coefficient is thus equal to the (N K)th order coefficient of a K (Remark ) We obtain this coefficient by differentiating N K K! ( x) K times ( x) K (26) and estimating the derivative at x 0 We obtain that K! a K K(K + ) (K + N K ) (N K)! a K (N )! K! (K )!(N K)! Thus, by combining (24) and (26), we obtain that the number of cluster of size n, when the N particles are distributed in K clusters, is (27) M n N,K (N n )!K!(N K)! (N )!(K 2)!(N n K + )!, which we remark to be independent of a The mean number of clusters of size n is obtained by summing over all possible configurations with K clusters, N (28) M n M n N,K Π K K (N n )! (N )! Using expression (95) for Π K, we obtain K K(K )(N K)! (N n K + )! Π K (29) M n 2a F ( N + + n; 2; 2a) if n < N, F ( N + ; 2; 2a) (30) and M N F ( N + ; 2; 2a) The mean number of clusters of size N is exactly equal to the probability Π (N) of having one cluster when there is N particles (see eq (96)) Indeed this is the only configuration where a cluster of size N can appear The mean number of clusters of size n is 2a Π (N) Π (N n), which means that it is given by the ratio of the probability of having one cluster when there are N particles over the probability of having one cluster when there are N n particles The number of clusters can also be written using the function G k defined in (03), (3) M n n ( ) k (2a)k+ n! (k + )! (n k)!k! G k k0 (32) n 2a ( ) k Π k+(n) Π (n) G k, k0

30 30 N HOZE AND D HOLCMAN where Π k (n) is the probability of having k clusters in a system of n particles To summarize this analysis, we plotted in Fig 5 the mean number of clusters of size n for N 5 particles A Number of clusters m <M > <M 2 > <M 3 > <M 4 > <M 5 > a B Number of clusters <Mn> a05 a5 a Cluster size n Fig 5 (A) Mean number of clusters of size n as a function of the parameter a for N 5 particles (equation (30)), and total number of clusters µ (B) Mean number of clusters M n as a function of the cluster size n, for a 005, a 05 and a 5 43 Probability to find two particles in the same cluster We now evaluate the probability to find two particles in the same cluster for a constant kernel When there are N particles, we first prove the following Lemma (33) where (34) N j M j N,K is given by relation (27) Proof:Using formula (27), we have (35) N j 2 M j N j j 2 M j N,K N + 2N N K K +, (N j )!K!(N K)! (N )!(K 2)!(N j K + )! N j j 2 (N j )!K!(N K)! (N )!(K 2)!(N j K + )! K(K )(N K)! (N )! N K+ j j 2 (N j )! (N j K + )!

31 STOCHASTIC COAGULATION-FRAGMENTATION 3 To determine the sum in eq (35), we introduce the sum (36) and prove now that (37) S N,K N K+ j j 2 (N j )! (N j K + )!, N! 2N K + S N,K (N K)! (K )K(K + ) We first obtain a recurrence relation between S N+,K and S N,K S N+,K where (39) and (40) A B N K+2 j N K+ j0 j 2 (N j)(n j)(n K j) (j + ) 2 (N j )(N j 2)(N K j) N K+ (N )! (N K + )! + (j 2 + 2j + )(N j )(N j 2)(N K j) j (N )! (N K + )! + S N,K + A + 2B, (38) N K+ j N K+ j (N j )(N j 2)(N K j) j(n j )(N j 2)(N K j) We first use a change of variable for A and find (4) A N 2 jk 2 N 2 jk 2 (K 2)! j(j )(j K + 3) j! (j K + 2)! N 2 jk 2 ( ) j K 2

32 32 N HOZE AND D HOLCMAN Using the binomial relation (42) n jk ( ) j k ( ) n +, k + we obtain (43) A ( ) N (K 2)! K (N )! (N K)!(K ) Similarly, we obtain for B (44) B N 2 jk 2 NA j! (N j ) (j K + 2)! N 2 jk 2 NA (K )! (j + )! (j K + 2)! N jk ( j ) K N! (N K)!(K ) N! (N K)!K N! (N K)!(K )K Finally we obtain the induction relation for S N,K N! S N+,K S N,K + 2 (N K)!(K )K + (N )! (N K)!(K ) (45) N!(2N K + 2) S N,K + (N K + )!(K )K + (N )! (N K + )!

33 STOCHASTIC COAGULATION-FRAGMENTATION 33 Using that S K,K (K 2)!, we finally evaluate the sum (46) S N,K (K )K 2 (K )K N jk N jk (j )!(2j K) (j K)! j! (j K)! (K )! N ( ) j 2(K 2)! (K 2)! K jk ( ) ( ) N + N 2(K 2)! (K 2)! K + K (K 2)! ( (N + )! 2 (N K)! (K + )! N! ) K! N! 2N K + (N K)! (K )K(K + ), N jk N jk (j )! (j K)! ( ) j K which is formula (33) Theorem 4 The probability to find two particles in the same cluster is (47) P 2 G, where G is defined in (02) Proof:Using eq (75) and lemma, we can now compute the probability to find two particles in the same cluster (48) Thus, P 2 N(N ) N(N ) N N Π K K j N K j 2 M j N,K N Π K ( N + 2N N K K + ) N (49) P 2 2 N N K Π K N K K + 2 N + 2N + N N K K + Π K

34 34 N HOZE AND D HOLCMAN Following formula (99), the sum in eq (49) can be expressed by integrating the Kummer s function N K + Π K Π z F ( N + ; 2; z)dz (50) 2 z 2 z2a K Integrating the hypergeometric series gives (5) z F ( N + ; 2; z)dz z 2 2F 2 ( N +, 2; 2, 3; z), which leads to (52) N K K + Π K 2F 2 ( N +, 2; 2, 3; 2a) 2 F ( N + ; 2; 2a) F ( N + ; 3; 2a) 2 F ( N + ; 2; 2a) By combining eq (49) and (52) we obtain (53) P 2 2 N + N + F ( N + ; 3; 2a) N F ( N + ; 2; 2a) The three-term recurrence relation for Kummer s function ([] Eqs ) gives F ( N + ; 3; 2a) N N + F ( N + 2; 3; 2a) + 2 N + F ( N + ; 2; 2a) (54) Finally, using eq (02), we obtain (55) P 2 G To finish this section, we note that the large N asymptotic of the probability that two particles are in the same cluster is (56) P 2 2 an Many results presented in this section can be used to study the distribution of clusters in biological systems such as telomere organization in yeast We provided here the explicit derivations of the exact and asymptotic formulas that can be used to analyze experimental and simulation results [4]

35 STOCHASTIC COAGULATION-FRAGMENTATION 35 5 Example 2: the case a i a for i < M and a i 0 if i M In this section, we consider N particles that can associate or dissociate at a constant rate, but in addition they cannot form clusters of more than M particles The configuration space for distributions of N particles in K clusters of size less than M is (57) P N,K,M {(m i ) i M ; M im i N, i M m i K} First, the minimal number of clusters is necessarily bounded by K N/M, since the opposite would imply a cluster of at least M + particles The probability of a configuration (m,, m M ) P N,K,M is equal to (58) P r{(m,, m M ) P N,K,M } i,m m!m M!, where the normalization constant,m is the Nth order coefficient of (59) (ax + ax ax M ) K a K X K ( XM X )K a K X ( X )K a K ( X) K K ( ) K ( ) n X nm n K ( ) K ( ) n X nm+k n Then the Nth order coefficient of the polynomial is obtained by finding the (N nm K)th order coefficient of ( X) K,M a K (60) K n0 n0 n0 ( ) ( K ( ) n n (N (nm + K))! D(N (nm+k)) ( X) K where we write D (n) (f) as the n th order derivative of some function f Thus, setting K 0 N K M, where is the floor function, we have (6),M a K K K 0 n0 (N nm )! n!(k n)!(n (nm + K))! ( )n For M N we find K 0 0 and the normalization constant (62),N a K (N )! (K )!(N K)!, ), X0

36 36 N HOZE AND D HOLCMAN is equal to the normalization constant obtained for the constant kernel in section 4 The mean number of clusters of size i M conditioned on the number of clusters K is M i K m i p(m, m M ) (63) m i P N,K,M a C N i,k,m,m To find the probability to have K clusters, we now redefine the formation rate In section 4, the formation rate was proportional to the number of pairs of particles since all of them could form a new cluster In the present case, two clusters of size i and j can form a new cluster only if i + j M The formation rate when there are K clusters is thus f K (64) (m i ) P N,K,M M/2 p(m,, m N ) i m i (m i ) 2 + M i,j i+j M;i j m i m j The formation rate can be written as a function of the coefficients,m as (65) f 2 C N,2,M, and for K > 2 (66) f K + K(K ) 2 K(K ) 2 min( M 2, N K+2 2 ) i min(m,n K+) i,j i+j M C N 2i,K 2,M C N i j,k 2,M The separation rate remains unchanged s K (N K)a, and the probabilities at steady state are given by (67) Π K f K+ s K Π K+ A simple expression is certainly hopeless, but the limit a 0 is informative: contrary to the previous case with a constant kernel (M N), where the

37 STOCHASTIC COAGULATION-FRAGMENTATION 37 particles form a single cluster of size N, the present system contains multiple steady state distributions The clusters grow independently and reach either their limit size M or are configured such that the sum of the sizes of each pair of cluster is larger than M All possible configurations contain exactly N/M clusters, where is the ceiling function We illustrate the limit case a 0 for N 9, M 4 (Fig 6) Because a > 0, all partitions are accessible, but as a 0, the steady state configurations are dominated by the configurations with the largest possible cluster size (4, 4, ), (4, 3, 2) and (3, 3, 3) Applying formulas (6) and (63), we obtain the limit cluster configuration probabilities (68) p(4, 4, ) 3 0 p(4, 3, 2) 6 0 p(3, 3, 3) 0 These steady state probabilities do not depend on the initial particles configurations as long as a 0 For a 0, there are three possible configurations (4, 4, ), (4, 3, 2) and (3, 3, 3): once equilibrium is attained, the clusters will remain unchanged The probability to get to equilibrium depends on the configuration and the order of clustering events When there is no limitation in the cluster formation (M N 9), a single cluster containing all particles is formed (Fig 6, left panel) For large values of a, most clusters are very small, and the distributions are similar for M 4 and M 9 (Fig 6, right panel) The probability for two particles to be in the same cluster provides a good estimation for the cluster distribution for various values of the parameter a (Fig 7) When a is large, most particles are contained in very small clusters and the probability P 2 is similar for the cases M 4 and M 9 When a 0, particles tend to form larger clusters A single cluster containing all particles is formed and P 2 when M 9, but the maximal value of P 2 is less than when the maximal cluster size is limited We can explicitly compute P 2 in the limit case a 0 For example for M 4, using eq (69), and summing over all possible configurations (68), we obtain P 2 p(4, 4, )P 2 (4, 4, ) + p(4, 3, 2)P 2 (4, 3, 2) + p(3, 3, 3)P 2 (3, 3, 3)